I looked and didn't understand

It seems interesting, but I've never gotten around to trying any of them. I might get P4G just so my Vita stops collecting dust.

Not really, 98% of the days in-game, you choose what activity you want to do. Build friendships, upgrade personal skills or fight. It's 100% story based, easy to follow and relatable. I dont wanna spoil the story or core gameplay elements for you.

The dungeon crawler elements remind me of Pikmin 2 except turn-based...

Really? I assumed the story was tied in because it's numbered. Are the characters continuous at all?

I never played Final Fantasy

Dear lord, how many good games did you miss out on ?

Do note that the dungeons in Persona are randomized ... 

Very few characters are recurring in the series so that's not a problem but the most important recurring character I'd say in Persona is Igor which is pretty much essential for every Persona protagonist's journey. 

Well, I've owned the following systems:

S Genesis - N64 - GC - Wii - Wii U - PS4 

GBC - GBA - DS - 3DS

As you can see, unless it's on Nintendo, I haven't played it yet

I just started to branch out...

Anyway, I'll look into it & see if I can start a game in the series sometime

Well, you don't have to play the other Persona games if you don't want to or it's too much trouble.

You can always start out with Persona 5 at next year on the PS4 to see how it's like ...

That's true. My OCD is bugging me that I'll start with 5 though.

If only it had another name...

If you want to start out that badly then get a PS2 and start with Persona 3: FES cause Persona 1 and 2 are outdated as hell ...

I can see you're a bit confused SOR, so allow me to give you a run down of everything you need to know. Sorry about any spelling mistakes, i ain't in the mood to proof read all this :p This is specifically about P4, but 3 follows similar ideas (there are no proper spoilers below, so don't worry):

P4 takes place over about a year. Each day you get the chance to partake in various activities, each of which will move time along specific amounts. Depending on the date, and how far along you've progressed in the story, you can do the follow:

- Social Links. You can spend time with various characters in the game to become closer to them, each of whom have their own little story that progresses as you level up their S.Link (up to level 10). When spending time with them you get prompted to choose how you want to react to a question or situation, and the actions you choose determine the spend you get closer to them. You can enter into a relationship with some characters, but it's not needed. Depending how close you are, some characters may also act in specific ways during battle (such as jumping in-front of a mortal attack to save you).
- Part-time Jobs. Pretty much what it says on the tin. Three of them are also linked to S.Links.
- Dungeons. These unlocked as the story progresses. You fight your way through each floor until you find the staircase to the next level, and continue this until you reach the boss. Each dungeon has its own unique twist (tied to specific characters), and some have unique mechanics.
- Attributes. You have 5 attributes (Courage, Knowledge, Expression, Understanding and Diligence). There are multiple way to level these up, from reading, jobs, studying, eating at a specific restraint, etc. These sometimes determine which S.Link choices you're aloud to make, as well as which Jobs you can take.
- Random shit. There's a lot of other side content i haven't mentioned, such as fishing, bug catching, going to the cinema with friends, a cafe, fortune draws, about 70 side quests, etc.

Combat:
- Persona. These are what you and your friends fight with. I can't really tell you what they (or 'Shadows') are without spoiling parts of the story, but they're essentially your primary weapon. You can sometimes win them after battle, or you can fuse Persona together to make stronger ones (or weaker one if you're so inclined :p). Any Persona you have bound to you when you leave a dungeon is added to the Compendium. This let's you re-buy them later on if you free one (since you can only hold a certain number at a time). Persona use up SP or HP points when attacking, and you can switch between any currently bound to you.
- As well as your Persona, you have a primary physical weapon (swords and the like) which use no HP or SP, items that you can buy in shops or find in chests, and gear that gives you various stat boosts (defense, evasion, elemental etc).
- 'Shadows'. In the dungeon corridors they look like black or red blobs, with their size being an indicator of if they are are below your level, the same, or above. If you touch them, you enter battle with whatever they are (you find out what monster they are once in battle). You are guaranteed to go first if you can hit one from behind, but the same goes for them if they manage to hit you from behind. If you hit them from the front no one gets a turn boost.

Let me know if you have any questions :p Also, hit me up if you ever get P4G. I can give you some spoiler free tips so that you don't miss the true ending or secret dungeon/epilogue (both of which are incredibly easy to miss).

One last thing; there's no need to play the main games in any specific order. They do take place in the same world, and are numbered chronologically, but outside of that they're unrelated. Most of their connection comes in the form of brief references or settings (for example in P4 you visit the school in which P3 took place).

Try not to think too much about the chronological story line of all the Persona games. It's a bit... messy ^^; If you really want to though, just read through the plot summaries on the wiki page of the ones you want to skip. Then read them 5 times more and decide it makes no damn scense :D